Step 1: Take the natural logarithm (ln) of both sides of the equation to simplify the expression and apply the logarithmic rule.

ln(y) = ln(x^x)

Step 2: Use the logarithmic rule ln(a^b) = b * ln(a) to simplify the expression further.

ln(y) = x * ln(x)

Step 3: Now, differentiate both sides of the equation with respect to x.

d/dx [ln(y)] = d/dx [x * ln(x)]

Step 4: Apply the chain rule on the left side and the product rule on the right side.

(d/dy [ln(y)]) * (dy/dx) = 1 * ln(x) + x * (d/dx [ln(x)])

Step 5: Since dy/dx is what we want to find, isolate it on the left side.

dy/dx = (1/ln(y)) * (ln(x) + x * (1/x))

Step 6: Simplify the expression further.

dy/dx = (1/ln(x^x)) * (ln(x) + 1)

Step 7: Finally, use the property ln(x^x) = x * ln(x) to get the final expression.

dy/dx = (1/x) * (ln(x) + 1)

So, the derivative of y = x^x with respect to x is dy/dx = (1/x) * (ln(x) + 1).